LE BOURGET, France—One of the biggest European players at this year’s Paris Air Show isn’t a European company at all. It’s General Electric Co.Over the past decade, the U.S. industrial icon has bought a number of aerospace companies and suppliers from Britain and Sweden to Italy and the Czech Republic. The deals have made GE parts ubiquitous on modern jetliners. It has grown into one of Europe’s largest aerospace employers, with roughly 11,600 workers across the continent, setting it apart from many of its peers.European deals are familiar turf for John Flannery, who last week was named GE’s new chief executive officer starting in August. He made his mark at the company leading GE’s biggest industrial acquisition ever, the $17 billion purchase of France’s Alstom SA power business.

GE is now trying to turn its collection of disparate European aerospace assets into a whole that is greater than the sum of its parts. It wants to knit the businesses into a pan-European group that can develop new equipment and tap European Union development funding.

“We pick the best companies at the time wherever they are globally,” David Joyce, GE vice chairman and boss of the aviation business said. “As long as it makes sense we buy them.”

Rivals are watching with interest. Eric Schulz, president for civil aerospace at Britain’s Rolls-Royce Holdings PLC, said “GE’s size has always been and I believe will always be a threat for Rolls-Royce and for the others, including their own customers.” The British aircraft-engine maker is GE’s biggest rival in powering the largest jetliners.

But Mr. Schulz said size has drawbacks, making companies less flexible as they try to protect broader corporate interest and, as a result, fail to offer the most competitive product. Being leaner “I hope gives us more creativity and innovation,” Mr. Schulz said. Rolls also is investing, including in the U.S., he said.

GE executives in Europe say that getting employees from a variety of companies and countries to learn the conglomerate’s ways and to cooperate is can be difficult, but their engineering talent benefits the group.

GE’s recent European spree is built on deep links. In 1941, GE built the U.S.’s first jet engine based on a British design. In 1974, it teamed up with a French rival, which was building engines for the supersonic Concorde, and today their CFM International joint venture is the world’s biggest producer of jet engines. In 2007, GE bought U.K. airplane-electronics maker Smiths Aerospace, expanding its aviation business beyond engines and landing gear.

The company’s massive plane-leasing business, GE Capital Aviation Services, known as GECAS, largely operates from Dublin and London. It has a fleet of about 1,700 planes with more than 200 customers. GECAS placed deals to buy 100 Airbus aircraft and another 20 from Boeing on the first day of the Paris Air Show unfolding outside the French capital this week.

When GE decided to expand into the civilian turboprop market, long dominated by U.S. rival Pratt & Whitney, a unit of United Technologies Corp. , it bought Prague-based Walter Aircraft Engines in 2008.

In 2013, GE snapped up Avio Aero, an important supplier near Milan whose private-equity owner was looking to exit. The company makes vital engines components for jets, turboprops and helicopters.

Last year, GE acquired 3-D printing companies in Germany and Sweden that are important suppliers to Avio, Walter and GE’s engine division in the U.S. Mr. Joyce said that when GE wanted to grow its 3-D activities, it had to strike deals in Europe because that is where the expertise resided.

Last week, on the day GE named Mr. Flannery its new boss, it also said it had bought OC Robotics of Bristol, England, for an undisclosed sum. The company’s snake-arm robots should make repairs in difficult to reach places on wings easier.

GE’s European empire “is something that came together almost accidentally,” said Riccardo Procacci, a GE veteran who moved to Avio from its Italian oil-and-gas business. “Stepping back now, we can see we are one of the biggest aviation companies in Europe.”

For GE, “a strong presence in Europe brings a different perspective on the world,” said Mr. Procacci.

Being based in Europe allows companies to sell locally developed high-tech components like engine combustors that in the U.S. face export restrictions due to national-security concerns, he said.

GE’s newly grown European roots give it access to national technology funding it couldn’t otherwise tap. In Italy, it has built a network of ties to universities and small business that benefit from government support.

And the various acquisitions bring GE a new continent of expertise it can draw from. When GE explored the best way to create a new turboprop engines, its experts in Prague and the U.S. looked at what other units, including Avio, could offer. “We said, this is going to be a European engine,” said Brad Mottier, a vice president at GE Aviation.